Construction and Analysis of Cryptographic Functions

verfasst von: Lilya Budaghyan

Verlag: Springer International Publishing

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Über dieses Buch

This book covers novel research on construction and analysis of optimal cryptographic functions such as almost perfect nonlinear (APN), almost bent (AB), planar and bent functions. These functions have optimal resistance to linear and/or differential attacks, which are the two most powerful attacks on symmetric cryptosystems. Besides cryptographic applications, these functions are significant in many branches of mathematics and information theory including coding theory, combinatorics, commutative algebra, finite geometry, sequence design and quantum information theory. The author analyzes equivalence relations for these functions and develops several new methods for construction of their infinite families. In addition, the book offers solutions to two longstanding open problems, including the problem on characterization of APN and AB functions via Boolean, and the problem on the relation between two classes of bent functions.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

In this chapter we give a short description of all the results presented in this monograph.

Lilya Budaghyan

2. Generalities

Abstract

In this chapter we present all necessary definitions and preliminary results on cryptographic functions which are used throughout this book.

Lilya Budaghyan

3. Equivalence Relations of Functions

Abstract

In this chapter we determine the cases where CCZ-equivalence coincide with EA-equivalence and where it is not, in particular, single and multioutput functions, bent functions, known power APN functions. We also study a possible extension of CCZ-equivalence.

Lilya Budaghyan

4. Bent Functions

Abstract

We construct new classes of nonquadratic bent Boolean and bent vectorial functions by applying CCZ-equivalence to a non-bent quadratic vectorial function F which has some bent components. We also solve an open problem proposed by Carlet, Charpin and Zinoviev in 1998 on characterization of APN and AB functions via Boolean functions, and a longstanding problem introduced by Dillon in 1974 about relation between two classes of bent functions.

Further we prove that many of the known classes of generalized bent functions do not intersect with the completed class of Maiorana-McFarland bent functions.

Lilya Budaghyan

5. New Classes of APN and AB Polynomials

Abstract

In this chapter we present several methods for construction of APN functions. Using these methods we construct 7 out of 11 known infinite families of quadratic APN polynomials CCZ-inequivalent to power functions, 4 of which are also AB when n is odd.

Lilya Budaghyan

6. Construction of Planar Functions

Abstract

We present new infinite families of perfect nonlinear quadratic multinomials over \({\mathbb{F}}_{p^{2k}}\) where p is any odd prime and k a positive integer. We prove that these families of planar functions define new commutative semifields. After the works of Dickson (1906) and Albert (1952), these were the firstly found infinite families of commutative semifields which are defined for all odd primes p.

One of these families has been constructed by extension of a known family of APN functions over \({\mathbb{F}}_{2^{2k}}\). This shows that known classes of APN functions over fields of even characteristic can serve as a source for further constructions of PN mappings over fields of odd characteristics.

Further we extend a known family of PN functions to a larger (up to CCZ-equivalence) family of PN functions by using isotopisms of semifields (which are not strong).

Lilya Budaghyan

Titel: Construction and Analysis of Cryptographic Functions
verfasst von: Lilya Budaghyan
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-12991-4
Print ISBN: 978-3-319-12990-7
DOI: https://doi.org/10.1007/978-3-319-12991-4